practice

Determine the following quantities in terms of the weight of the bridge span.

The weight of the counterweight and the tension when the span is down.

The tension when the bridge is open and the torque needed to keep the span up.

Write something completely different.

conceptual

A hammer is balanced on one finger as shown in the diagram to the right. This divides the hammer up into two parts: the handle on the left and the head (plus a little bit of the handle) on the right. Which part weighs more — or are they both equally heavy? Justify your answer.

How can all three types of equilibrium — stable, unstable, and neutral — be demonstrated using…

an egg

a cone

a torus (a shape like a donut or a bagel)

Explain the reasoning behind each of the following general rules of design.

numerical

A 1.50 m tall woman lies on a light (massless) board that is supported by two scales, one under her feet and one under her head. The two scales read 320 N and 294 N, respectively. How far is the center of gravity of her body from her feet?

statistical

center-of-population.txtThe center of population of the United States as defined by the Census Bureau is the same as that of the center of gravity of a collection of point masses on a plane. It is the point at which a weightless, smooth, spherical shell in the shape of the "lower 48" states and the District of Columbia would balance if weights of identical size were placed on it — each weight representing the location of one person. On such an imaginary surface, north-south distances between parallels of latitude (ϕ) are identical and their angular measure in degrees may be used as units of displacement. In contrast, east-west distances between meridians of longitude (λ) are not constant but vary with latitude from a maximum at the equator to zero at the poles.
Multiplying by the cosine of the latitude will correct for this convergence of the meridians at the poles. In addition, small areas of the country are used as data points rather than individual human beings, which reduces the computational burden. (In 1960 43,000 areas were used but by 2000, this number had risen to more than 8,000,000. By 2020 or 2030, the number of areas will probably equal the number of residents.)

Thus, the center of population of the US computed by the Census Bureau is the point whose latitude (φ̅) and longitude (λ̅) satisfy the equations…

φ̅ =

∑ wiφi

λ̅ =

∑ wiλicos φi

∑ wi

∑ wicos φi

Where ϕi, λi, and wi are the latitude, longitude, and population of the census areas included in the calculation.

The data on the accompanying tabs-delimited text file give the population and the effective latitude and longitude in degrees of the fifty states and the District of Columbia from the 2000 census. Using this data, determine the coordinates of the population center of the United States at this time. (Be sure to exclude Alaska and Hawaii from your calculations, but do include the District of Columbia.)

In what state is this point located? Which county? What is the nearest incorporated community (city, village, or town)? What is the nearest street intersection?

Go to this location and await further instructions.

investigative

Determine the mass of a ruler using a known weight to balance it over a pivot.